The terminology I was taught back in the day, when I used to sell barware commercially:
DOF, AKA "Double Old-Fashioned," or "rocks" glass = short, squatty heavy-bottomed glass, used for things like Gimlets, Rusty Nails, Vodka rocks, Scotch rocks, and such - in other words, strong drinks served over ice
Highball, what you all are calling a "tumbler" = a taller version of a rocks glass, used for drinks like Collinses, G&T's, Rum & Cokes, Whisky & sodas, and such
Cocktail glass, AKA "Martini" glass = for shaken & strained drinks, not just Martinis (such as the quaint old Pink Squirrels and Grasshoppers, and newfangled stuff like Cosmopolitans)
Snifters = for neat drinks, either room temp or shaken & strained unmixed drinks
Cooler = a taller, narrower highball glass, used for nasty things like Long Island Iced Teas and sometimes Mai Tais, depending on how much foo-foo is perched on the rim and inserted in the drink